Differential Equations TUTORIAL For Lecture and for Practical Exercise
Abstract
The textbook is compiled in accordance with the program of the discipline “Differential Equations” (Section - Ordinary Differential Equations). Basic theoretical information and 115 examples of solving typical problems are given, as well as recommendations for studying the discipline. Examples are provided for independent work, 85 examples and 20 questions for self-test, recommended literature.This collection was prepared in accordance with the undergraduate curriculum (Construction) in the subject “Differential Equations” in the Republic of Uzbekistan based on existing state educational standards. 60730500 – Design and operation of water supply and sewerage systems, 60730800 – Road construction (by type of activity), 60710400 – Ecology and environmental protection (by industry sector), 60722500 – Geodesy, cartography and cadastre (by function), 60722800 – Cadastre (by type of activity), 60730300 – Civil engineering: construction of buildings and structures, 60730400 – Construction and installation of engineering communications (according to types) provided for by the undergraduate curriculum, Prepared according to the scientific program. Here, the method of successive approximations of solutions to differential equations is considered, the theorem for the existence of a solution to a differential equation and the uniqueness theorem are proved. It includes theoretical information on practical subjects in the field of mathematical sciences, exemplary samples and practical problems for independent solution.
Downloads
Metrics
References
2. Benchohra, M., Bouriah, S., Salim, A. & Zhou, Y. (2024). Fractional Differential Equations: A Coincidence Degree Approach. Berlin, Boston: De Gruyter. https://doi.org/10.1515/9783111334387
3. Sun, Z., Zhang, Q. & Gao, G. (2023). Finite Difference Methods for Nonlinear Evolution Equations. Berlin, Boston: De Gruyter. https://doi.org/10.1515/9783110796018
4. Lasalle, J. (Ed.). (2012). International symposium on nonlinear differential equations and nonlinear mechanics. Elsevier.
5. Prabowo PS, Mungkasi S. A multistage successive approximation method for Riccati differential equations. Bull Electr Eng Inform. 2021; 10(3): 1589-1597. doi: 10.11591/eei.v10i3.3043
6. Hoyos FE, Candelo JE, Taborda JA. Selection and validation of mathematical models of power converters using rapid modeling and control prototyping methods. Int J Electr Comput Eng. 2018; 8(3): 1551. doi: 10.11591/ijece.v8i3.pp1551-1568
7. Schuch D. Nonlinear Riccati equations as a unifying link between linear quantum mechanics and other fields of physics. J Phys Conf Ser. 2014; 504(1):012005. doi:10.1088/1742-6596/504/1/012005
8. Dehghan M, Shakeri F. Approximate solution of a differential equation arising in astrophysics using the variational iteration method. New Astron. 2008; 13: 53-59. doi:10.1016/j.newast.2007.06.012
9. Geng F, Lin Y, Cui M. A piecewise variational iteration method for Riccati differential equations. Comput Math Appl. 2009; 58: 2518-2522. doi:10.1016/j.camwa.2009.03.063
10. Ghorbani A, Momani S. An effective variational iteration algorithm for solving Riccati differential equations. Appl Math Lett. 2010; 23: 922-927. doi:10.1016/j.aml.2010.04.012
11. Abbasbandy S. A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials. J Comput Appl Math. 2007; 207: 59-63. doi:10.1016/j.cam.2006.07.012
12. Geng F. A modified variational iteration method for solving Riccati differential equations. Comput Math Appl. 2010; 60: 1868-1872. doi:10.1016/j.camwa.2010.07.017
13. Abbasbandy S. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method. Appl Math Comput. 2006; 172: 485-490. doi:10.1016/j.amc.2005.02.014
14. Abbasbandy S. Iterated He's homotopy perturbation method for quadratic Riccati differential equation. Appl Math Comput. 2006; 175: 581-589. doi:10.1016/j.amc.2005.07.035
15. Bulut H, Evans DJ. On the solution of the Riccati equation by the decomposition method. Int J Comput Math. 2002; 79: 103-109. doi:10.1080/00207160211917
16. Gümgüm S, Baykuş-Savaşaneril N, Kürkçü ÖK, Sezer M. Lucas polynomial approach for second order nonlinear differential equations. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi. 2020; 24(1): 230-236. doi:10.19113/sdufenbed.546847
17. Yüzbaşı Ş. A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differential–difference equations. Comput Math Appl. 2012; 64(6): 1691-1705. doi:10.1016/j.camwa.2012.01.026
18. Chen D, Wang Q, Li Y, Li Y, Zhou H, Fan Y. A general linear free energy relationship for predicting partition coefficients of neutral organic compounds. Chemosphere. 2020; 247:125869. doi:10.1016/j.chemosphere.2020.125869
19. Mehmood A, Zameer A, Ling SH, Rehman AU, Raja MAZ. Integrated computational intelligent paradigm for nonlinear electric circuit models using neural networks, genetic algorithms and sequential quadratic programming. Neural Comput & Applic. 2020; 32: 10337-10357. doi:10.1007/s00521-019-04573-3
20. Xie X, Huang L, Marson SM, Wei G. Emergency response process for sudden rainstorm and flooding: scenario deduction and Bayesian network analysis using evidence theory and knowledge meta-theory. Nat Hazards. 2023; 1-23: 3307-3329. doi:10.1007/s11069-023-05988-x
21. Piskunov N. S. "Differential and integral calculus." Moscow “Science” - 1985 2 – part
22. Shchipachev V.S. “Higher Mathematics”. Moscow “Higher School” 1985.
23. Danko P.E., Popov. A.G., Kozhevnikova T.Ya. "Higher mathematics in exercises and problems. Part -2. " Moscow "Higher School" 1986.
24. Minorsky V.P. “Collection of problems in higher mathematics” Moscow “Science” 1987.
25. Glagolev A.A., Solntseva T.V. "Course of Higher Mathematics." Publishing house "Higher School" Moscow -1971.
26. Bronshtein I.N., Semendyaev K.A. "Handbook of mathematics" for engineers and college students. Moscow "Science" 1986.
27. Natanson I.P. “A short course in higher mathematics” Publishing house “Nauka” Moscow 1968.
28. Myshkis A.D. “Lectures on higher mathematics” Publishing house “Nauka” Moscow 1967.
29. Ignatieva V., Krasnoshchekova T.I., Smirnov V.F. "Course of Higher Mathematics" Ed. "Higher School" Moscow 1968.
30. Soatov Y.U. “Higher mathematics” part - 4. Tashkent “Ukituvchi” 1998.
31. Muxtarov Ya., Soleyev A. “Ordinary differential equations examples and problems. Textbook. Samarkand – 2020.
32. Khusanov B., Zikryeyayev Sh., Muxtarov Ya. “Hidher mathematics” (differential equations) tutorial. Samarkand – 2020.
33. Y.U. Soatov. “Higher mathematics” part - 2. Tashkent “Ukituvchi” 1998.
Copyright (c) 2024 Kulmirzayeva Gulrabo Abduganiyevna
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors retain copyright and grant the Umsida Press right of first publication with the work simultaneously licensed under a Creative Commons Attribution 4.0 International License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this platform.
